Question

Let $R$ and $S$ be two equivalence relations on a set $A$. Then

• A. $R\cup S$ is an equivalence relation on $A$
• B. $R\cap S$ is an equivalence relation on $A$
• C. $R-S$ is an equivalence relation on $A$
• D. None of the above

Answer 1

The correct option is B

$R\cap S$ is an equivalence relation on $A$

Explanation for the correct option:

Since $R$ and $S$ are reflexive this means for any $a\in A$

$(a,a)\in R$ and $(a,a)\in S$

This means $(a,a)\in R\cap S$

Therefore $R\cap S$is reflexive

Consider $\left(a,b\right)\in R\cap S$

Then $(a,b)\in R\left(a,b\right)\in S$

Now, since $R$ and $S$ are symmetric $(b,a)\in R$ and $(b,a)\in S$

Therefore $(b,a)\in R\cap S$

Therefore $R\cap S$is symmetric

Now, consider $(a,b),(b,c)\in R\cap S$

This means $(a,b),(b,c)\in R$

Hence $(a,c)\in R$ since $R$ is transitive

And $(a,b),(b,c)\in S$

This means $(a,c)\in S$ since $S$is transitive

Therefore $(a,c)\in R\cap S$

hence $R\cap S$is transitive

Therefore $R\cap S$is an equivalence relation on $A$

Hence, option (B) is the correct answer